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Bayesian estimation of discretely observed multi-dimensional diffusion processes using guided proposals

机译:使用指导性建议的离散观测多维扩散过程的贝叶斯估计

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Estimation of parameters of a diffusion based on discrete time observations poses a difficult problem due to the lack of a closed form expression for the likelihood. From a Bayesian computational perspective it can be casted as a missing data problem where the diffusion bridges in between discrete-time observations are missing. The computational problem can then be dealt with using a Markov-chain Monte-Carlo method known as data-augmentation. If unknown parameters appear in the diffusion coefficient, direct implementation of data-augmentation results in a Markov chain that is reducible. Furthermore, data-augmentation requires efficient sampling of diffusion bridges, which can be difficult, especially in the multidimensional case. We present a general framework to deal with with these problems that does not rely on discretisation. The construction generalises previous approaches and sheds light on the assumptions necessary to make these approaches work. We define a random-walk type Metropolis-Hastings sampler for updating diffusion bridges. Our methods are illustrated using guided proposals for sampling diffusion bridges. These are Markov processes obtained by adding a guiding term to the drift of the diffusion. We give general guidelines on the construction of these proposals and introduce a time change and scaling of the guided proposal that reduces discretisation error. Numerical examples demonstrate the performance of our methods.
机译:基于离散时间观测值的扩散参数估计提出了一个难题,因为缺乏针对可能性的闭合形式表达式。从贝叶斯计算的角度来看,可以将其视为丢失数据问题,其中离散时间观测值之间的扩散桥丢失。然后可以使用称为数据增强的马尔可夫链蒙特卡罗方法处理计算问题。如果在扩散系数中出现未知参数,则直接执行数据增强会导致可还原的马尔可夫链。此外,数据扩充需要对扩散桥进行有效采样,这可能很困难,尤其是在多维情况下。我们提出了一个不依赖离散化处理这些问题的通用框架。该构造概括了先前的方法,并阐明了使这些方法起作用所需的假设。我们定义了一个随机游走型Metropolis-Hastings采样器来更新扩散桥。我们的方法是使用指导性建议对扩散桥进行采样来说明的。这些是通过向扩散的漂移添加指导项而获得的马尔可夫过程。我们为这些建议的构建提供了一般指导,并介绍了随时间变化和指导性建议的缩放比例,以减少离散化误差。数值示例证明了我们方法的性能。

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